Theorem unisngl 21330

Description: Taking the union of the set of singletons recovers the initial set. (Contributed by Thierry Arnoux, 9-Jan-2020.)

Hypothesis

Ref	Expression
dissnref.c	|- C = { u | E. x e. X u = { x } }

Assertion

Ref	Expression
unisngl	|- X = U. C

Distinct variable groups:   u, C, x   u, X, x

Proof of Theorem unisngl

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dissnref.c	. . 3 |- C = { u | E. x e. X u = { x } }
2	1	unieqi 4445	. 2 |- U. C = U. { u | E. x e. X u = { x } }
3		simpl 473	. . . . . . . . 9 |- ( ( y e. u /\ u = { x } ) -> y e. u )
4		simpr 477	. . . . . . . . 9 |- ( ( y e. u /\ u = { x } ) -> u = { x } )
5	3, 4	eleqtrd 2703	. . . . . . . 8 |- ( ( y e. u /\ u = { x } ) -> y e. { x } )
6	5	exlimiv 1858	. . . . . . 7 |- ( E. u ( y e. u /\ u = { x } ) -> y e. { x } )
7		eqid 2622	. . . . . . . 8 |- { x } = { x }
8		snex 4908	. . . . . . . . 9 |- { x } e. _V
9		eleq2 2690	. . . . . . . . . 10 |- ( u = { x } -> ( y e. u <-> y e. { x } ) )
10		eqeq1 2626	. . . . . . . . . 10 |- ( u = { x } -> ( u = { x } <-> { x } = { x } ) )
11	9, 10	anbi12d 747	. . . . . . . . 9 |- ( u = { x } -> ( ( y e. u /\ u = { x } ) <-> ( y e. { x } /\ { x } = { x } ) ) )
12	8, 11	spcev 3300	. . . . . . . 8 |- ( ( y e. { x } /\ { x } = { x } ) -> E. u ( y e. u /\ u = { x } ) )
13	7, 12	mpan2 707	. . . . . . 7 |- ( y e. { x } -> E. u ( y e. u /\ u = { x } ) )
14	6, 13	impbii 199	. . . . . 6 |- ( E. u ( y e. u /\ u = { x } ) <-> y e. { x } )
15		velsn 4193	. . . . . 6 |- ( y e. { x } <-> y = x )
16		equcom 1945	. . . . . 6 |- ( y = x <-> x = y )
17	14, 15, 16	3bitri 286	. . . . 5 |- ( E. u ( y e. u /\ u = { x } ) <-> x = y )
18	17	rexbii 3041	. . . 4 |- ( E. x e. X E. u ( y e. u /\ u = { x } ) <-> E. x e. X x = y )
19		r19.42v 3092	. . . . . 6 |- ( E. x e. X ( y e. u /\ u = { x } ) <-> ( y e. u /\ E. x e. X u = { x } ) )
20	19	exbii 1774	. . . . 5 |- ( E. u E. x e. X ( y e. u /\ u = { x } ) <-> E. u ( y e. u /\ E. x e. X u = { x } ) )
21		rexcom4 3225	. . . . 5 |- ( E. x e. X E. u ( y e. u /\ u = { x } ) <-> E. u E. x e. X ( y e. u /\ u = { x } ) )
22		eluniab 4447	. . . . 5 |- ( y e. U. { u | E. x e. X u = { x } } <-> E. u ( y e. u /\ E. x e. X u = { x } ) )
23	20, 21, 22	3bitr4ri 293	. . . 4 |- ( y e. U. { u | E. x e. X u = { x } } <-> E. x e. X E. u ( y e. u /\ u = { x } ) )
24		risset 3062	. . . 4 |- ( y e. X <-> E. x e. X x = y )
25	18, 23, 24	3bitr4i 292	. . 3 |- ( y e. U. { u | E. x e. X u = { x } } <-> y e. X )
26	25	eqriv 2619	. 2 |- U. { u | E. x e. X u = { x } } = X
27	2, 26	eqtr2i 2645	1 |- X = U. C

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   {csn 4177   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by:  dissnref  21331  dissnlocfin  21332